\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 133, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/133\hfil On exact solutions]
{On exact solutions of a degenerate quasilinear wave equation with source term}

\author[N.-E. Amroun, A. Benaissa\hfil EJDE-2006/133\hfilneg]
{Nour-Eddine Amroun, Abb\`es Benaissa}  % in alphabetical order


\address{Universit\'e Djillali Liab\`es \\
Faculty des Sciences, Departement de mathematiques \\
B. P. 89, Sidi Bel Abb\`es 22000, Algeria}
\email[N.-E. Amroun]{amroun\_nour@yahoo.com}
\email[A. Benaissa]{benaissa\_abbes@yahoo.com}

\date{}
\thanks{Submitted June 21, 2006. Published October 19, 2006.}
\subjclass[2000]{35C05, 35L70}
\keywords{Kirchhoff equation; exact solutions}

\begin{abstract}
 In this paper, we construct exact solutions of the Cauchy problem
 of degenerate quasilinear wave equation of Kirchhoff type, and
 study the effect of the nonlinear terms on the existence  of
 solutions. We construct solutions that exist globally for some
 initial data, and  that  blow up in a finite time for some
 other initial data.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]

\section{Introduction}

In this paper, we construct exact solutions of the following
initial-value problem of a quasilinear wave equation with nonlinear
source term, and  study their asymptotic behaviour concerning
the time variable,
\begin{equation} \label{eP}
\begin{gathered}
(|u'|^{l-2}u')'-  M(\|u_{x}\|_{2}^{2}) u_{xx} =\gamma |u|^{p-1}u,
\quad \hbox{in } \mathbb{R}\times [0, +\infty[,\\
u(x, 0)=u_{0}(x),\quad u'(x, 0)=u_{1}(x)\hbox{in }\mathbb{R}.
\end{gathered}
\end{equation}
where $M(r)=r^{\frac{p-1}{2}}$,
$\|u_{x}\|_{2}^{2}=\int_{-\infty}^{\infty}|u_{x}(x, t)|^{2}\, dx$,
$l> 2$, $p> l-1$ and $\gamma> 0$ are constants.

For  problem \eqref{eP}, when $l> 2$ and $M\equiv 1$ without source
term and with nonlinear dissipation, Benaissa and Mimouni {\cite{bemi}}
determined suitable relations between $l$ and $p$, so that the energy
decays exponentially or alternatively polynomially.
More precisely, they showed that the energy of the solutions decays with
exponentially if $l+1\geq p$ and decays polynomially if $l+1< p$.

For problem \eqref{eP} (in the case of bounded domain), when $l> 2$ and $M$
is not a constant function, with nonlinear
dissipation, Benaissa and Messaoudi {\cite{beme}} have investigated
the blowup of solutions. They show that, for suitably chosen initial
data and a relation between $l$ and $p$, any classical solution blows up
in finite time.

When $l=2$, the equation without a source term is often called the
wave equation of Kirchhoff type which has been introduced
for studying  nonlinear vibrations of an
elastic string by Kirchhoff \cite{kirc}. The existence of local
and global solutions in Sobolev and Gevrey classes was investigated by many
authors; see for example \cite{kaya,gome2,kaji}.

When $l=2$,   Ebihara, Hoshino and Kurokiba \cite{ebih}  constructed
one of the solutions for \eqref{eP} and studied their behavior
in $t$. They have shown that there exists a solution which blows
up at a finite time under some initial condition
and in the other case there exists a global solution which decays with
the order $O(t^{-\frac{2}{p-1}})(t\to \infty)$.
We think that the interaction of the term $(|u'|^{l-2}u')'$ ($l> 2$) and
the source term $|u|^{p-1}u$ and the velocity $M(r)$ have an effect on the
result of \cite{ebih}.

We construct the solution of the form  $u(x, t)=v(x)\varphi(t)$ for
\eqref{eP} when
$M(r)=r^{\frac{p-1}{2}},p>1, l> 2$ and $p+3=2l$ and we study the
behavior of the solutions as the time $t$ increases.

\section{Separation of variables}

Let $u(x, t)=v(x)\varphi(t)$ and substitute in \eqref{eP}.
Then  \eqref{eP} is changed to
\begin{equation} \label{eP'}
\begin{gathered}
\begin{aligned}
&|v|^{l-2}v (|\varphi'|^{l-2}\varphi')'
-\Big(\int_{-\infty}^{\infty}|v_{x}(x, t)|^{2}\, dx\Big)^{\frac{p-1}{2}}\\
&\times |\varphi|^{p-1}\varphi v_{xx}-\gamma |v|^{p-1}v |\varphi|^{p-1}\varphi=0,
\quad  x\in \mathbb{R}, \quad t> 0,
\end{aligned}\\
u(x, 0)=v(x)\varphi(0)=\varphi_{0}v(x), \quad x\in \mathbb{R}, \\
u_{t}(x, 0)=v(x)\varphi_{t}(0)=\varphi_{1}v(x), \quad x\in \mathbb{R}.
\end{gathered}
\end{equation}
Here we consider the case $\varphi(t)> 0$, so that the first equation
of \eqref{eP'} is equivalent to
\begin{equation}
\frac{\alpha^{p-1} v_{xx}+\gamma |v|^{p-1}v}{|v|^{l-2}v}
=\frac{(|\varphi'|^{l-2}\varphi')'}{\varphi^{p}}=\lambda,
\label{e1}
\end{equation}
where $\alpha^{2}=\int_{-\infty}^{\infty}|v_{x}(x)|^{2}\, dx$ and
$\lambda$ is a positive constant.
Therefore, we obtain two problems from (\ref{e1}): First problem
\begin{equation} \label{eP1}
\begin{gathered}
\alpha^{p-1} v_{xx}-\lambda |v|^{l-2}v+\gamma |v|^{p-1}v=0,\quad
x\in\mathbb{R}, \\
\alpha^{2}=\int_{-\infty}^{\infty}|v_{x}|^{2}\, dx< +\infty\,,
\end{gathered}
\end{equation}
and second problem
\begin{equation} \label{eP22}
\begin{gathered}
(|\varphi'|^{l-2}\varphi')'=\lambda \varphi^{p}, \quad  t\geq 0, \\
\varphi(0)=\varphi_{0}, \quad \varphi_{t}(0)=\varphi_{1},\\
\varphi(t)\geq 0,\quad  t\geq 0.
\end{gathered}
\end{equation}

\section{First Problem}

In this section, we construct a positive solution $v(x)$ of
\eqref{eP1} with $\lim_{|x|\to \infty}v(x)=0$.
For this purpose we study  the problem
\begin{equation} \label{eP1'}
\begin{gathered}
\alpha^{p-1} v_{xx}-\lambda |v|^{l-2}v+\gamma |v|^{p-1}v=0,\quad x\in\mathbb{R}, \\
v(0)=A\quad  (\hbox{a positive constant}), \\
v_{x}(0)=0, \\
\lim_{|x|\to \infty} v(x)=0, \\
\frac{\alpha^{2}}{2}=\int_{0}^{\infty}|v_{x}(x)|^{2}\, dx< +\infty,
\end{gathered}
\end{equation}
where $\alpha$ is a fixed positive number. If $v(x)$ is a solution
of \eqref{eP1'}, then we can solve
\eqref{eP1} by setting $v(-x)=v(x)$ for $x> 0$, because of the second
equation in \eqref{eP1'}.
Multiplying the first equation of \eqref{eP1'} by $2 v_{x}$ and integrating
from $0$ to $x$, we obtain
\begin{equation}
\alpha^{p-1} v_{x}^{2}=\frac{2\lambda}{l} v^{l}-\frac{2\gamma}{p+1}v^{p+1}-
\frac{2\lambda}{l} A^{l}+\frac{2\gamma}{p+1}A^{p+1}.
\label{e2}
\end{equation}
If we choose $A> 0$ such that
\begin{equation}
A=\Big(\frac{(p+1)\lambda}{\gamma\ l}\Big)^{\frac{1}{p+1-l}},
\label{e3}
\end{equation}
then (\ref{e2}) implies
\begin{equation}
\alpha^{\frac{p-1}{2}} v_{x}
=\mp \sqrt{\frac{2\lambda}{l} v^{l}-\frac{2\gamma}{p+1}v^{p+1}}.
\label{e4}
\end{equation}
Here we consider the case where $v$ is positive and $v_{x}<0$,
so that we treat the following
equation which is derived from (\ref{e4}):
\begin{equation}
\frac{v_{x}}{v^{\frac{l}{2}}\sqrt{\frac{2\lambda}{l}-\frac{2\gamma}{p+1}v^{p+1-l}}}=-\alpha^{-\frac{p-1}{2}}.
\label{e5}
\end{equation}
If we integrate (\ref{e5}) from $c$ to $v$, then we obtain
\begin{equation}
\int_{v}^{c}\frac{dz}{z^{\frac{l}{2}}\sqrt{\frac{2\lambda}{l}
-\frac{2\gamma}{p+1}z^{p+1-l}}}=\alpha^{-\frac{p-1}{2}}x.
\label{e6}
\end{equation}
If there exists $x^{*}\in (0, \infty)$ such that $v(x^{*})=0$, then we get
$$
\int_{0}^{c}\frac{dz}{z^{\frac{l}{2}}\sqrt{\frac{2\lambda}{l}-\frac{2\gamma}{p+1}z^{p+1-l}}}=\alpha^{-\frac{p-1}{2}}x^{*}.
$$
But one can easily show
$\int_{0}^{c}\frac{dz}{z^{\frac{l}{2}}\sqrt{\frac{2\lambda}{l}
-\frac{2\gamma}{p+1}z^{p+1-l}}}=\infty$
with use of (\ref{e3}) and $\alpha^{-\frac{p-1}{2}} x^{*}< +\infty$, thus
$v(x)$ is monotone decreasing and $v(x)> 0$. And if
$\lim_{x\to \infty}v(x)=k> 0$, then  from (\ref{e6}), we obtain
$$
\int_{k}^{c}\frac{dz}{z^{\frac{l}{2}}\sqrt{\frac{2\lambda}{l}-\frac{2\gamma}{p+1}z^{p+1-l}}}
=\lim_{x\to\infty}\alpha^{-\frac{p-1}{2}}x\quad (=\infty).
$$
However, $\int_{k}^{c}\frac{dz}{z^{\frac{l}{2}}\sqrt{\frac{2\lambda}{l}-\frac{2\gamma}{p+1}z^{p+1-l}}}$
is finite, so we deduce that $\lim_{x\to\infty}v(x)=0$.

By putting $y=\sqrt{\frac{2\lambda}{l}-\frac{2\gamma}{p+1}z^{p+1-l}}$
in order to calculate the left hand side
of (\ref{e6}), we see that
$$
I=\int_{v}^{c}\frac{dz}{z^{\frac{l}{2}}\sqrt{\frac{2\lambda}{l}
-\frac{2\gamma}{p+1}z^{p+1-l}}}
=\int_{0}^{\sqrt{\frac{2\lambda}{l}-\frac{2\gamma}{p+1}z^{p+1-l}}}
\frac{2^{\frac{2p- l}{2(p+1-l)}}(\frac{\gamma}{(p+1)})^{\frac{l-2}{2(p+1-l)}}}{(p+1-l)(\frac{2\lambda}{l}-y^{2})^{\frac{2p- l}{2(p+1-l)}}}
\, dy.
$$
We suppose that $\frac{2p-l}{2(p+1-l)}=\frac{3}{2}$. We obtain
$$
I=\int_{0}^{\sqrt{\frac{2\lambda}{l}-\frac{2\gamma}{p+1}z^{l-2}}}
\frac{2^{\frac{3}{2}}(\frac{\gamma}{(p+1)})^{\frac{1}{2}}}{(l-2)(\frac{2\lambda}{l}-y^{2})^{\frac{3}{2}}}\, dy
= \frac{l}{\lambda(l-2)}\frac{\sqrt{\frac{2\lambda}{l}-\frac{2\gamma}{p+1}v^{l-2}}}{v^{\frac{l-2}{2}}}.
$$
Then (\ref{e6}) becomes
$$
\frac{l}{\lambda(l-2)}\frac{\sqrt{\frac{2\lambda}{l}-\frac{2\gamma}{p+1}v^{l-2}}}{v^{\frac{l-2}{2}}}=
\alpha^{-\frac{p-1}{2}}x,
$$
which implies
\begin{equation}
v(x)=\Big(\frac{l/(2\lambda)}
{\big(\frac{\lambda(l-2)}{l}\big)^{2}\alpha^{-(p-1)}x^{2}
+\frac{2\gamma}{p+1}}\Big)^{1/(l-2)}.
\label{e7}
\end{equation}
We put $\mu=\big(\frac{\lambda(m-2)}{m}\big)^{2}\alpha^{-(p-1)}$ and
differentiate (\ref{e7}), to obtain
$$
v_{x}(x)=-\frac{2}{m-2}\big(\frac{l}{2\lambda}\big)^{\frac{1}{l-2}} \mu
\frac{x}{\big(\mu\ x^{2}+\frac{2\gamma}{p+1}\big)^{1+\frac{1}{l-2}}}.
$$
Thus we have
$$
\int_{0}^{\infty}|v_{x}(x)|^{2}\, dx
=\Big(\frac{2}{l-2}\left(\frac{m}{2\lambda}\right)^{\frac{1}{l-2}}\Big)^{2}
\mu^{2}\int_{0}^{\infty}\frac{x^{2}}
{\big(\mu\ x^{2}+\frac{2\gamma}{p+1}\big)^{2+\frac{2}{l-2}}}\, dx.
$$
When we put
$H=\int_{0}^{\infty} \frac{x^{2}}{\big(\mu\ x^{2}+\frac{2\gamma}{p+1}
\big)^{2+\frac{2}{l-2}}}\, dx$
and let $z=\sqrt{\mu /a}\ x$, where $a=\frac{2\gamma}{p+1}$, we have
\begin{align*}
H&=\frac{1}{a^{\frac{1}{2}
+\frac{2}{l-2}} \mu^{\frac{3}{2}}}\int_{0}^{\infty}
\frac{z^{2}}{(1+z^{2})^{2+\frac{2}{l-2}}} \, dz\\
&= \frac{l-2}{2l} \frac{1}{a^{\frac{1}{2}
  +\frac{2}{l-2}} \mu^{\frac{3}{2}}}\int_{0}^{\infty}
\frac{1}{(1+z^{2})^{1+\frac{2}{l-2}}} \, dz\hfill \\
&= \frac{l-2}{2l} \frac{1}{a^{\frac{1}{2}+\frac{2}{l-2}}
\mu^{\frac{3}{2}}}K\quad  (K< +\infty).
\end{align*}
Therefore,
$$
\int_{0}^{\infty}|v_{x}(x)|^{2}\, dx=
\left(\frac{2}{l-2}\left(\frac{m}{2\lambda}\right)^{\frac{1}{l-2}}\right)^{2}
\sqrt{\mu} \frac{l-2}{2 l} \frac{1}{a^{\frac{1}{2}+\frac{2}{l-2}} } K\,.
$$
If we take $\alpha$ such as
$$
\alpha=\big\{\frac{4\sqrt{2}}{\sqrt{l}}\big(\frac{l(l-1)}{2}
\frac{\gamma}{\lambda}\big)^{\frac{1}{2}+\frac{2}{l-2}}K \big\}^{1/l}
\lambda^{\frac{3}{2 l}},
$$
then \eqref{eP1'}(4) is satisfied. Therefore, we have the following
theorem.

\begin{theorem} \label{thm1}
If $l> 2$ and $p=2 l- 3$, then the solution $v(x)$ of \eqref{eP1'} such that
$v(0)=\big(\frac{2(l-2)}{l}\frac{\lambda}{\gamma}\big)^{\frac{1}{l-2}}$
and $v_{x}(0)=0$, is given by
$$
v(x)=\Big(\frac{1/(2\lambda)}
{\big(\frac{\lambda(l-2)}{l}\big)^{2}\alpha^{-(p-1)}x^{2}
+\frac{2\gamma}{p+1}}\Big)^{1/(l-2)},
$$
where
$$
\alpha=\big\{\frac{4\sqrt{2}}{\sqrt{l}}
\big(\frac{l(l-1)}{2}\frac{\gamma}{\lambda}\big)^{\frac{1}{2}+\frac{2}{l-2}}K
\big\}^{1/l} \lambda^{\frac{3}{2 l}},\quad
K=\int_{0}^{\infty}\frac{1}{(1+z^{2})^{1+\frac{2}{l-2}}}.
$$
\end{theorem}

This solution can be extended to a solution  of \eqref{eP1}.

\section{Second Problem}

In this section, we consider Problem
\begin{equation} \label{eP2}
\begin{gathered}
(|\varphi'|^{l-2}\varphi')'=\lambda \varphi^{p},\quad t\geq 0, \\
\varphi(0)=\varphi_{0},\quad \varphi_{t}(0)=\varphi_{1}, \\
\varphi(t)\geq 0,\quad t\geq 0
\end{gathered}
\end{equation}
where $\lambda$ is the constant from the previous section.
If we multiply the equation \eqref{eP2}(1) by $\varphi_{t}(t)$ and
integrate from $0$ to $t$ as in section $2$, then we have
\begin{equation}
\varphi_{t}(t)=\pm\Big(\frac{l\lambda}{(l-1)(p+1)}v^{p+1}
+|\varphi_{1}|^{l}- \frac{l\lambda}{(l-1)(p+1)}\varphi_{0}^{p+1}
\Big)^{1/l}, \label{e8}
\end{equation}
because of \eqref{eP2}(2) and \eqref{eP2}(3).
In order that we construct the solution of \eqref{eP2} from (\ref{e8}),
the following condition has to be satisfied
$$
G(\varphi)\equiv\frac{l\lambda}{(l-1)(p+1)}v^{p+1}+|\varphi_{1}|^{l}
- \frac{l\lambda}{(l-1)(p+1)}\varphi_{0}^{p+1}\geq 0.
$$
We consider the following three cases.

\noindent \textbf{Case 1.}
If $\varphi_{0}\geq 0$ and $\varphi_{1}> 0$, then
$G(\varphi(0))=G(\varphi_{0})> 0$. Hence
$\varphi_{t}(t)=(G(\varphi(t)))^{1/l}> 0$ for
sufficiently small $t> 0$ since $(G(\varphi(t)))^{1/l}$
is monotone increasing function, we see that $\varphi_{t}> 0$
for all $t> 0$ where $\varphi(t)$ exist. Since one can show
$\int_{\varphi_{0}}^{\infty}(1\backslash (G(\varphi))^{1/l})\,
d\varphi< +\infty$, we see
that $\varphi\to +\infty$ as $t\to T^{*}$, where
$T^{*}=\int_{\varphi_{0}}^{\infty}(1\backslash (G(\varphi))^{1/l})\,
d\varphi$.

\noindent \textbf{Case 2.}
In the case of $\varphi_{0}\geq 0$ and
$|\varphi_{1}|^{l}- \frac{l\lambda}{(l-1)(p+1)}\varphi_{0}^{p+1}= 0$,
by solving (\ref{e8}), we obtain
$$
\varphi_{\pm}(t)=\Big(\varphi_{0}^{-\frac{p+1-l}{l}}\mp \frac{(p+1-l)
\big(\frac{\lambda l}{(l-1)(p+1)}\big)^{1/l}}{l}t
\Big)^{-\frac{l}{p+1-l}},
$$
with double signs in same order. Obviously $\varphi_{+}$  decays with the
order $\mathcal{O}(t)^{-\frac{l}{p+1-l}}$ as
$t\to +\infty$ and $\varphi_{-}$ blows up at
$\varphi_{0}^{-\frac{p+1-l}{l}}
\frac{l}{p+1-l} \big(\frac{(l-1)(p+1)}{\lambda l}\big)^{1/l} $.

\noindent \textbf{Case 3.}
If $\varphi_{0}\geq 0, \varphi_{1}< 0$ and
$|\varphi_{1}|^{l}- \frac{l\lambda}{(l-1)(p+1)}\varphi_{0}^{p+1}> 0$,
we have $\varphi_{t}(t)=-(G(\varphi(t)))^{1/l}< 0$ locally. Then, since
$G(\varphi(t))$ is monotone increasing function, we see that
$\varphi_{t}(t)< 0$ for all $t> 0$ where $\varphi(t)$ exist.
From (\ref{e8}),
$$
\int_{\varphi_{0}}^{0}-\frac{d\varphi}{(G(\varphi))^{1/l}}\equiv T^{**}< +\infty.
$$
Therefore, $T^{**}$ exists such that $\varphi(t)\to 0$ as $t\to T^{**}$.
From (\ref{e8}), we can get
\begin{equation}
\int_{\varphi(t)}^{0}-\frac{1}{T^{**}-t}\frac{d\varphi}{(G(\varphi))^{1/l}}=1.
\label{e9}
\end{equation}
Let $s=(T^{**}-t)r$ in (\ref{e9}) and let $t\to T^{**}$, we have
\begin{equation}
\lim_{t\to T^{**}}\varphi(t)(T^{**}-t)^{-1}
=\Big(|\varphi_{1}|^{l}- \frac{l\lambda}{(l-1)(p+1)}\varphi_{0}^{p+1}
\Big)^{1/l}. \label{e10}
\end{equation}
From (\ref{e8}) and (\ref{e10}), we have
\begin{equation}
\lim_{t\to T^{**}}\varphi(t)\varphi'(t)(T^{**}-t)^{-1}
=-\Big(|\varphi_{1}|^{l}- \frac{l\lambda}{(l-1)(p+1)}\varphi_{0}^{p+1}
\Big)^{2/l}.
\label{e11}
\end{equation}
Then we have the following theorem.

\begin{theorem} \label{thm2}
{\bf (1)} If $\varphi_{0}\geq 0$ and $\varphi_{1}> 0$, then by putting
$$
T^{*}=\int_{\varphi_{0}}^{\infty}(1\backslash (G(\varphi))^{1/l})\,
d\varphi,
$$
the solution $\varphi(t)$ of \eqref{eP2} blows up at $t=T^{*}$, that
is $\varphi(t)\to +\infty$ as $t\to T^{*}$.

\noindent
{\bf (2)} If $\varphi_{0}\geq 0$ and $|\varphi_{1}|^{l}
- \frac{l\lambda}{(l-1)(p+1)}\varphi_{0}^{p+1}= 0$, then the solution
$\varphi(t)$ of \eqref{eP2} is given by
$$
\varphi_{\pm}(t)=\Big(\varphi_{0}^{-\frac{p+1-l}{l}}\mp
\frac{(p+1-l)\big(\frac{\lambda l}{(l-1)(p+1)}\big)^{1/l}}{l} t
\Big)^{-\frac{l}{p+1-l}},
$$
with double signs in the same order.

\noindent
{\bf (3)} If $\varphi_{0}\geq 0, \varphi_{1}< 0$ and $|\varphi_{1}|^{l}
- \frac{l\lambda}{(l-1)(p+1)}\varphi_{0}^{p+1}> 0$, then by putting
$$
\int_{\varphi_{0}}^{0}-\frac{d\varphi}{(G(\varphi))^{1/l}}
\equiv T^{**},
$$
the solution $\varphi(t)$ vanishes at $t=T^{**}$ and satisfies
(\ref{e10}) and (\ref{e11}).

Depending on the choice of the constants $\varphi_{i}\ (i=0, 1)$,
we obtain a solution which blows up in a finite time and
another case we have a solution which decays with order
$O(t)^{-\frac{l}{p+1-l}}$ as $t\to \infty$.
\end{theorem}


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\end{document}